Cycle conjectures and birational invariants over finite fields

TL;DR

This paper explores a birational invariant over finite fields and demonstrates its deep connections to major conjectures like Tate, Beilinson, and Grothendieck--Serre, showing that their validity in some degrees implies their validity in all degrees.

Contribution

It introduces a natural birational invariant for varieties over finite fields and establishes its equivalence to key conjectures, revealing new implications across degrees.

Findings

Vanishing of the invariant on projective space links to major conjectures.

Partial validity of conjectures in some degrees implies their validity in all degrees.

The invariant provides a new perspective on the relationships between these conjectures.

Abstract

We study a natural birational invariant for varieties over finite fields and show that its vanishing on projective space is equivalent to the Tate conjecture, the Beilinson conjecture, and the Grothendieck--Serre semi-simplicity conjecture for all smooth projective varieties over finite fields. We further show that the Tate, Beilinson, and 1-semi-simplicity conjecture in half of the degrees implies those conjectures in all degrees.